How can I specify filenames of animation renders as [SCENE NAME] + [FRAME] when rendering an image sequence?
This answer already explains how to render certain frames.

bpy.context.scene.name             ...   the scene's name
bpy.context.scene.frame_start      ...   the scene's starting frame




import bpy

scene = bpy.context.scene
fp = scene.render.filepath # get existing output path

for frame_nr in range(scene.frame_start, scene.frame_end):

    # set current frame
    scene.frame_set(frame_nr)

    # set output path
    scene.render.filepath = "//render/" + scene.name + "_folder/" + scene.name + "_customName_" + '{0:03d}'.format(frame_nr)
    bpy.ops.render.render(write_still=True)    
 scene.render.filepath = fp


Refer to:


padding ints with python
the range function